Molecular geometry and chain entanglement: parameters for the tube model

The tube diameter in the reptation model is the distance between a given chain segment and its nearest segment in adjacent chains. This dimention is thus related to the cross-sectional area of polymer chains and the nearest approach among chains, without effects of thermal fluctuation and steric repulsion. Prior calculated tube diameters are much larger, about 5 times, than the actual chain cross-sectional areas. This is ascribed to the local freedom required for mutual rearrangement among neighboring chain segments. This tube diameter concept seems to us to infer a relationship to the corresponding entanglement spacing. Indeed, we report here that the critical molecular weight, M_c, for the onset of entanglements is found to be M_c = 28 A/(〈R²〉₀/M), where A is the chain cross-sectional area and 〈R²〉₀ the mean-square end-to-end distance of a freely jointed chain of molecular weight M. The new, computed relationship between the critical number of backbone atoms for entanglement and the chain cross-sectional area of polymers, N_c = A^0,44, is concordant with the cross-sectional area of polymer chains being the parameter controlling the critical entanglement number of backbone atoms of flexible polymers.